Details

The volume of this 3D figure is . We look for the maximum value of for . By , we get as an approximate solution and .

According to the referee of the Journal Origami Society of Japan, maximizing the volume of a cup made from a square sheet of paper is a new problem proposed by the authors. Our previous research is presented in [2].

The origami wrapping problem, which has been studied by some mathematicians, looks for the biggest object that can be wrapped with a sheet of paper. See [1].

References

[1] M. L. Catalano-Johnson and D. Loeb, "Problem 10716: A Cubical Gift," American Mathematical Monthly, 108(1), 2001 pp. 81–82.

[2] W. Ogasa, T. Inoue, K. Nishimura, T. Nakaoka, D. Ikeda, A. Kanno, S. Nakamura, H. Matsui, T. Yamauchi, S. Utsuki, and R. Miyadera, "The Maximization of a Cup Made from a Square Sheet of Paper," Science of Origami, 1(1), 2011, in Japanese.